-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/
-- | Compensated floating-point arithmetic
--
-- This package provides compensated floating point arithmetic.
@package compensated
@version 0.2
-- | This module provides a fairly extensive API for compensated floating
-- point arithmetic based on Knuth's error free transformation, various
-- algorithms by Ogita, Rump and Oishi, Hida, Li and Bailey, Kahan
-- summation, etc. with custom compensated arithmetic circuits to do
-- multiplication, division, etc. of compensated numbers.
--
-- In general if a has x bits of significand, Compensated
-- a gives you twice that. You can iterate this construction for
-- arbitrary precision.
--
-- References:
--
--
module Numeric.Compensated
class (RealFrac a, Precise a, Floating a) => Compensable a where data family Compensated a
with :: (Compensable a, Compensable a) => Compensated a -> (a -> a -> r) -> r
compensated :: (Compensable a, Compensable a) => a -> a -> Compensated a
magic :: Compensable a => a
-- | This provides the isomorphism between the compact representation we
-- store these in internally and the naive pair of the primal and
-- residual components.
_Compensated :: Compensable a => Iso' (Compensated a) (a, a)
type Overcompensated a = Compensated (Compensated a)
-- | This Lens lets us edit the primal directly, leaving the
-- residual untouched.
primal :: Compensable a => Lens' (Compensated a) a
-- | This Lens lets us edit the residual directly, leaving
-- the primal untouched.
residual :: Compensable a => Lens' (Compensated a) a
-- | Extract the primal component of a compensated value,
-- when and if compensation is no longer required.
uncompensated :: Compensable a => Compensated a -> a
-- | fadd a b k computes k x y such that
--
--
-- x + y = a + b
-- x = fl(a + b)
--
--
-- but only under the assumption that abs a >=
-- abs b. If you aren't sure, use add.
--
-- Which is to say that x is the floating point image of (a
-- + b) and y stores the residual error term.
fadd :: Num a => a -> a -> (a -> a -> r) -> r
-- | add a b k computes k x y such that
--
--
-- x + y = a + b
-- x = fl(a + b)
--
--
-- Which is to say that x is the floating point image of (a
-- + b) and y stores the residual error term.
add :: Num a => a -> a -> (a -> a -> r) -> r
-- | times a b k computes k x y such that
--
--
-- x + y = a * b
-- x = fl(a * b)
--
--
-- Which is to say that x is the floating point image of (a
-- * b) and y stores the residual error term.
--
-- This could be nicer if we had access to a hardware fused multiply-add.
times :: Compensable a => a -> a -> (a -> a -> r) -> r
-- | squared a k computes k x y such that
--
--
-- x + y = a * a
-- x = fl(a * a)
--
--
-- Which is to say that x is the floating point image of (a
-- * a) and y stores the residual error term.
squared :: Compensable a => a -> (a -> a -> r) -> r
divide :: Compensable a => a -> a -> (a -> a -> r) -> r
-- | error-free split of a floating point number into two parts.
--
-- Note: these parts do not satisfy the compensated contract
split :: Compensable a => a -> (a -> a -> r) -> r
-- | Perform Kahan summation over a list.
kahan :: (Foldable f, Compensable a) => f a -> Compensated a
-- | Calculate a scalar + compensated sum with Kahan summation.
(+^) :: Compensable a => a -> Compensated a -> Compensated a
-- | Compute a * Compensated a
(*^) :: Compensable a => a -> Compensated a -> Compensated a
-- | Calculate a fast square of a compensated number.
square :: Compensable a => Compensated a -> Compensated a
instance (Compensable a, Precise a) => Precise (Compensated a)
instance Compensable a => Floating (Compensated a)
instance (Compensable a, Unbox a) => Vector Vector (Compensated a)
instance (Compensable a, Unbox a) => MVector MVector (Compensated a)
instance (Compensable a, Storable a) => Storable (Compensated a)
instance (Compensable a, Serialize a) => SafeCopy (Compensated a)
instance (Compensable a, Serialize a) => Serialize (Compensated a)
instance (Compensable a, Binary a) => Binary (Compensated a)
instance Compensable a => RealFrac (Compensated a)
instance Compensable a => Real (Compensated a)
instance Compensable a => Fractional (Compensated a)
instance Compensable a => Enum (Compensated a)
instance Compensable a => Num (Compensated a)
instance (Reviewable p, Functor f, Compensable a, a ~ b) => Snoc p f (Compensated a) (Compensated b) a b
instance (Reviewable p, Functor f, Compensable a, a ~ b) => Cons p f (Compensated a) (Compensated b) a b
instance Compensable a => Monoid (Compensated a)
instance Compensable a => Semigroup (Compensated a)
instance Compensable a => Ord (Compensated a)
instance Compensable a => Eq (Compensated a)
instance (Applicative f, Compensable a, Compensable b) => Each f (Compensated a) (Compensated b) a b
instance (Compensable a, Read a) => Read (Compensated a)
instance (Compensable a, Show a) => Show (Compensated a)
instance (Compensable a, Data a) => Data (Compensated a)
instance (Compensable a, Hashable a) => Hashable (Compensated a)
instance Typeable1 Compensated
instance Compensable a => Compensable (Compensated a)
instance Compensable Float
instance Compensable Double